Distribution of existential quantifiers, with a bound-variable
hypothesis saying that is not free in , but can
be
free in
(and there is no distinct variable condition on and
). (Contributed
by Mario Carneiro, 20-Mar-2013.)

Specialization, using implicit substitution. Compare Lemma 14 of
[Tarski] p. 70. The spim1915
series of theorems requires that only one
direction of the substitution hypothesis hold. (Contributed by NM,
5-Aug-1993.) (Revised by Mario Carneiro, 3-Oct-2016.)

A variable introduction law for equality. Lemma 15 of [Monk2] p. 109,
however we do not require to be distinct from and
(making the proof longer). (Contributed by NM, 5-Aug-1993.) (Proof
shortened by Andrew Salmon, 25-May-2011.)

Derive ax-11o2080 from a hypothesis in the form of ax-111715. ax-102079 and
ax-111715 are used by the proof, but not ax-10o2078 or ax-11o2080. TODO:
figure out if this is useful, or if it should be simplified or
eliminated. (Contributed by NM, 2-Feb-2007.)

When the class variables in definition df-clel2279 are replaced with set
variables, this theorem of predicate calculus is the result. This
theorem provides part of the justification for the consistency of that
definition, which "overloads" the set variables in wel1685
with the class
variables in wcel1684. Note: This proof is referenced on the
Metamath
Proof Explorer Home Page and shouldn't be changed. (Contributed by NM,
28-Jan-2004.) (Proof modification is discouraged.)

When the class variables in definition df-clel2279 are replaced with set
variables, this theorem of predicate calculus is the result. This
theorem provides part of the justification for the consistency of that
definition, which "overloads" the set variables in wel1685
with the class
variables in wcel1684. (Contributed by NM, 28-Jan-2004.) (Revised
by
Mario Carneiro, 21-Dec-2016.)

This theorem can be used to eliminate a distinct variable restriction on
and and replace it with the
"distinctor"
as an antecedent. normally has free and can be read
, and
substitutes for and can be read
. We don't require that and be
distinct: if
they aren't, the distinctor will become false (in multiple-element
domains of discourse) and "protect" the consequent.

To obtain a closed-theorem form of this inference, prefix the hypotheses
with , conjoin them, and apply dvelimdf2022.

Note that is a
dummy variable introduced in the proof. On the web
page, it is implicitly assumed to be distinct from all other variables.
(This is made explicit in the database file set.mm). Its purpose is to
satisfy the distinct variable requirements of dveel21960 and ax-171603. By
the end of the proof it has vanished, and the final theorem has no
distinct variable requirements. (Contributed by NM, 29-Jun-1995.)
(Proof modification is discouraged.)

The specialization axiom of standard predicate calculus. It states that
if a statement holds for all , then it also holds for the
specific case of
(properly) substituted for . Translated to
traditional notation, it can be read: ",
provided that is
free for in ."
Axiom 4 of
[Mendelson] p. 69. See also spsbc3003 and rspsbc3069. (Contributed by NM,
5-Aug-1993.)